A Fenchel-Moreau theorem for ${\bar{L}}^0$-valued functions

Abstract

We establish a Fenchel-Moreau type theorem for proper convex functions $f:X\to {\bar{L}}^0$, where $(X,Y,⟨\cdot ,\cdot ⟩)$ is a dual pair of Banach spaces and ${\bar{L}}^0$ is the space of all extended real-valued functions on a $\sigma$-finite measure space. We introduce the concept of stable lower semi-continuity which is shown to be equivalent to the existence of a dual representation $$f(x)=\underset{y\in L^0(Y)}{\sup }\left\{⟨x,y⟩-f^{\ast }(y)\right\},x\in X,$$ where $L^0(Y)$ is the space of all strongly measurable functions with values in $Y$, and $⟨\cdot ,\cdot ⟩$ is understood pointwise almost everywhere. The proof is based on a conditional extension result and conditional functional analysis.

10 pages, [v2]: incorporating reviewer's comments, final version accepted for publication

References in corpus (4)

• The algebra of conditional sets and the concepts of conditional topology and compactness
• Lagrange Duality in Set Optimization
• Versions of Eberlein-Šmulian and Amir-Lindenstrauss theorems in the framework of conditional sets
• A perturbational duality approach in vector optimization

Cited by in corpus (1)

• On compactness in $L^0$-modules